PROPOSITION A. THEOREM.

If the first of four magnitudes has the same ratio to the second, which the third has to the fourth; then, if the first be greater than the second, the third is also greater than the fourth; and if equal, equal; if less, less.

Take any equimultiples of each of them, as the doubles of each: then, by def. 5th of this book, if the double of the first be greater than the double of the second, the double of the third is greater than the double of the fourth:

but if the first be greater than the second,

the double of the first is greater than the double of the second; wherefore also the double of the third is greater than the double of the fourth;

therefore the third is greater than the fourth:

in like manner if the first be equal to the second, or less than it, the third can be proved to be equal to the fourth, or less than it. Therefore, if the first, &c. Q.E.D.

PROPOSITION B. THEOREM.

If four magnitudes are proportionals, they are proportionals also when taken inversely.

A

G

Let A be to B, as Cis to D.

Then also inversely, B shall be to A, as D to C.

Take of B and D any equimultiples whatever E and F;
and of A and C any equimultiples whatever G and H.
First, let E be greater than G, then G is less than E:
and because A is to B, as C is to D, (hyp.)

and of A and C, the first and third, G and H are equimultiples; and of B and D, the second and fourth, E and F are equimultiples; and that G is less than E, therefore H is less than F; (v. def. 5.) that is, Fis greater than H;

if, therefore, E be greater than G,
Fis greater than H;

in like manner, if E be equal to G,
F may be shewn to be equal to H;
and if less, less;

but E, F, are any equimultiples whatever of B and D, (constr.)
and G, H any whatever of A and C';
therefore, as B is to A, so is D to C. (v. def. 5.)
Therefore, if four magnitudes, &c, Q. E.D.

PROPOSITION C. THEOREM.

If the first be the same multiple of the second, or the same part of it, that the third is of the fourth; the first is to the second, as the third is to the fourth.

Let the first A be the same multiple of the second B,

that the third Cis of the fourth D.

Take of A and C any equimultiples whatever E and F; and of B and D any equimultiples whatever G and H. Then, because A is the same multiple of B that Cis of D; (hyp.) and that E is the same multiple of A, that Fis of C; (constr.) therefore E is the same multiple of B, that Fis of D; (v. 3.) that is, E and F are equimultiples of B and D: but G and H are equimultiples of B and D; (constr.) therefore, if E be a greater multiple of B than G is of B. Fis a greater multiple of D than H is of D; that is, if E be greater than G, Fis greater than H:

in like manner, if E be equal to G, or less than it,
F may be shewn to be equal to H, or less than it,
but E, Fare equimultiples, any whatever, of A, C; (constr.)
and G, H any equimultiples whatever of B, D;

therefore A is to B, as C is to D. (v. def. 5.) Next, let the first A be the same part of the second B, that the third Cis of the fourth D.

A

Then A shall be to B, as Cis to D.

For since A is the same part of B that Cis of D, therefore B is the same multiple of A, that D is of C: wherefore, by the preceding case, B is to A, as D is to C; and therefore inversely, A is to B, as Cis to D. (V. B.) Therefore, if the first be the same multiple, &c.

PROPOSITION D. THEOREM.

Q. E.D.

If the first be to the second as the third to the fourth, and if the first be a multiple, or a part of the second; the third is the same multiple, or the same part of the fourth.

Let A be to B as Cis to D:
and first, let A be a multiple of B.
Then C shall be the same multiple of D.

Take E equal to A,

D

F

and whatever multiple A or E ís of B, make F the same multiple of D:

then, because A is to B, as Cis to D; (hyp.)

and of B the second, and D the fourth, equimultiples have been taken, E and F;

therefore A is to E, as C to F: (v. 4. Cor.)

but A is equal to E, (constr.)

therefore C is equal to F: (V. A.)

and Fis the same multiple of D, that A is of B; (constr.)
therefore C is the same multiple of D, that A is of B.
Next, let A the first be a part of B the second.

Then C the third shall be the same part of D the fourth.
Because A is to B, as Cis to D; (hyp.)

A

then, inversely, B is to A, as D to C: (v. B.)

B

C

D

but A is a part of B, therefore B is a multiple of A: (hyp.) therefore, by the preceding case, D is the same multiple of C; that is, Cis the same part of D, that A is of B.

Therefore, if the first, &c.

Q. E.D.

PROPOSITION VII. THEOREM.

Equal magnitudes have the same ratio to the same magnitude: and the same has the same ratio to equal magnitudes.

Let A and B be equal magnitudes, and C any other. Then A and B shall each of them have the same ratio to C: and C shall have the same ratio to each of the magnitudes A and B.

Take of A and B any equimultiples whatever D and E,
and of Cany multiple whatever F.

Then, because D is the same multiple of A, that E is of B, (constr.)
and that A is equal to B: (hyp.)
therefore D is equal to E; (v. ax. 1.)

therefore, if D be greater than F, E is greater than F;
and if equal, equal; if less, less:

but D, E are any equimultiples of A, B, (constr.)
and Fis any multiple of C;

therefore, as A is to C, so is B to C. (v. def. 5.)
Likewise C shall have the same ratio to A, that it has to B.
For having made the same construction,

D may

in like manner be shewn to be equal to E;
therefore, if F be greater than D,

it is likewise greater than E;
and if equal, equal; if less, less;
but Fis any multiple whatever of C,

and D, E are any equimultiples whatever of A, B;
therefore, Cis to A as Cis to B. (v. def. 5.)
Therefore, equal magnitudes, &c. Q.E.D.

Of two unequal magnitudes, the greater has a greater ratio to any other magnitude than the less has and the same magnitude has a greater ratio to the less of two other magnitudes, than it has to the greater.

Let AB, BC be two unequal magnitudes, of which AB is the greater, and let D be any other magnitude.

Then AB shall have a greater ratio to D than BC has to D:
and D shall have a greater ratio to BC than it has to AB.

If the magnitude which is not the greater of the two AC, CB, be not less than D.

take EF, FG, the doubles of AC, CB, (as in fig. 1.)

but if that which is not the greater of the two AC, CB, be less than D.
(as in fig. 2 and 3.) this magnitude can be multiplied, so as to
become greater than D, whether it be AC, or CB.
Let it be multiplied until it become greater than D,
and let the other be multiplied as often;

and let EF be the multiple thus taken of AC,
and FG the same multiple of CB:

therefore EF and FG are each of them greater than D:
and in every one of the cases,

take the double of D, K its triple, and so on,

till the multiple of D be that which first becomes greater than FG: let Z be that multiple of D which is first greater than FG, and K the multiple of D which is next less than L.

Then because L is the multiple of D, which is the first that becomes greater than FG,

the next preceding multiple K is not greater than FG;

that is, FG is not less than K:

and since EF is the same multiple of AC, that FG is of CB; (constr.) therefore FG is the same multiple of CB, that EG is of AB; (v. 1.) that is, EG and FG are equimultiples of AB and CB; and since it was shewn, that FG is not less than K, and, by the construction, EF is greater than D; therefore the whole EG is greater than K and D together: but K together with D is equal to L; (constr.) therefore EG is greater than L;

but FG is not greater than L: (constr.)

and EG, FG were proved to be equimultiples of AB, BC;
and I is a multiple of D; (constr.)

therefore AB has to D a greater ratio than BC has to D. (v. def. 7.) Also D shall have to BC a greater ratio than it has to AB.

it

For having made the same construction,
may be shewn in like manner, that I is greater than FG,
but that it is not greater than EG:

and L is a multiple of D; (constr.)

and FG, EG were proved to be equimultiples of CB, AB: therefore D has to CB a greater ratio than it has to AB. (v. def. 7.) Wherefore, of two unequal magnitudes, &c. Q. E. D.

PROPOSITION IX. THEOREM.

Magnitudes which have the same ratio to the same magnitude are equal to one another: and those to which the same magnitude has the same ratio are equal to one another.

Let, A, B have each of them the same ratio to C.
Then A shall be equal to B.

For, if they are not equal, one of them must be greater than the other: let A be the greater:

then, by what was shewn in the preceding proposition,

there are some equimultiples of A and B, and some multiple of C, such,
that the multiple of A is greater than the multiple of C,
but the multiple of B is not greater than that of C,
let these multiples be taken;

and let D, E be the equimultiples of A, B,
and F the multiple of C,

such that D may be greater than F, but E not greater than F.
Then, because A is to Cas B is to C, (hyp.)
and of A, B, are taken equimultiples, D, E,
and of C is taken a multiple F;

and that D is greater than F;

therefore E is also greater than F: (v. def. 5.)

but E is not greater than F; (constr.) which is impossible: therefore A and B are not unequal; that is, they are equal. Next, let C have the same ratio to each of the magnitudes  and B. Then A shall be equal to B.

For, if they are not equal, one of them must be greater than the other: let A be the greater:

therefore, as was shewn in Prop. VIII.
there is some multiple F of C,

and some equimultiples E and D, of B and A such,
that Fis greater than E, but not greater than D:
and because C is to B, as C is to 4, (hyp.)

and that F the multiple of the first, is greater than E the multiple of the second;

therefore F the multiple of the third, is greater than D the multiple of the fourth: (v. def. 5.)